Problem 28 - Advanced Engineering Mathematics

Question

The distance from \(\pi i\) to 0 is \(|\pi i|=\pi\)

Step-by-Step Solution

Answer

The distance from \( \pi i \) to 0 is \( \pi \).

Step 1: Understanding the Problem

The problem requires us to find the distance from the point represented by the complex number \( \pi i \) to the origin, which is the point 0, in the complex plane.

Step 2: Identify the Formula

To find the distance of a complex number \( a + bi \) from the origin, we use the modulus, calculated as \( \sqrt{a^2 + b^2} \). In this case, the complex number is \( 0 + \pi i \).

Step 3: Apply the Formula

For \( \pi i \), the real part \( a = 0 \) and the imaginary part \( b = \pi \). The modulus is \( \sqrt{0^2 + \pi^2} = \sqrt{\pi^2} \).

Step 4: Simplify the Expression

We simplify \( \sqrt{\pi^2} \) to get \( \pi \). Therefore, the distance from \( \pi i \) to 0 is \( \pi \).

Key Concepts

Modulus of a Complex Number

In mathematics, the modulus of a complex number is a fundamental concept used to measure the distance from the complex number to the origin on the complex plane. When dealing with a complex number represented as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part, the modulus is calculated using the formula:\[|a + bi| = \sqrt{a^2 + b^2}\]For example, in our original problem with the complex number \(\pi i\), we see it as \(0 + \pi i\). Here, \(a = 0\) and \(b = \pi\). Thus, the modulus becomes \(\sqrt{0^2 + \pi^2} = \pi\).
The modulus provides us with a way to grasp the 'size' or 'magnitude' of a complex number, similar to how we perceive the length of a real number on the number line.

Complex Plane

The complex plane, sometimes known as the Argand plane, is a graphical representation of complex numbers. This plane consists of two perpendicular axes: the horizontal real axis and the vertical imaginary axis. Each complex number \(a + bi\) corresponds to a unique point in this plane.
For instance, the complex number \(\pi i\) is not just represented by a coordinate -- it's a point directly above the origin (0, 0) on the imaginary axis, at a distance \(\pi\) units upwards.

• The real part (\(a\)) of a number indicates how far the point is from the origin along the horizontal axis.
• The imaginary part (\(b\)) specifies the vertical distance from the origin.

Understanding the complex plane helps us visualize complex numbers, providing a clearer insight into their nature and relationships. This concept is also critical for operations like addition, subtraction, and determining distances between points, such as finding the modulus.

Imaginary Numbers

Imaginary numbers are numbers that can be written as a real number multiplied by the imaginary unit \(i\), where \(i\) is defined by the property \(i^2 = -1\). This concept might initially seem abstract, but it's incredibly useful in mathematics and engineering.
Consider the original problem featuring the number \(\pi i\). Here, \(\pi\) is the real number coefficient of the imaginary unit \(i\). Imaginary numbers extend the number line concept from one dimension into two, forming a plane.

• These numbers allow for meaningful solutions to equations that don't have real roots such as \(x^2 + 1 = 0\).
• They facilitate complex analysis and are crucial in signal processing and control theory.

By introducing the imaginary unit, calculations involving complex numbers become quite direct. Imaginary numbers work hand-in-hand with real numbers to form complex numbers, enriching the numerical landscape.